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Algebraic properties of homotopy classes in homotopy theory

Research Project

Project/Area Number 14540063
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionChiba University

Principal Investigator

MARUYAMA Kenichi  Chiba University, Faculty of Education, Associate Professor, 教育学部, 助教授 (70173961)

Co-Investigator(Kenkyū-buntansha) KOSHIKAWA Hiroaki  Chiba University, Faculty of Education, Professor, 教育学部, 教授 (60000866)
YAMAUCHI Kenichi  Chiba University, Faculty of Education, Professor, 教育学部, 教授 (20009690)
TSUKIYAMA Kouzou  Shimane University, Faculty of Education, Professor, 教育学部, 教授 (20093651)
Project Period (FY) 2002 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsAlgebraic topology / Homotopy theory / Homotopy sets / Automorphism groups / Nilpotent groups / 自己ホモトピー集合
Research Abstract

Homotopy classes of maps of ten have binary operations. It is very useful to study their algebraic properties to know the topology of spaces. Homotopy groups and (co) homology groups are typical examples of such algebraic structures.
In this project we dealt with the self-homotopy sets of spaces. In this case homotopy sets are monoid by the binary operation induced by composition of maps. It is well known that studying composition of maps is very important in homotopy theory. Here mainly we study two kinds of subsets of self-homotopy sets. First we consider natural subgroups of self-homotopy sets which consists of maps inducing the trivial map on homotopy. Secondly we study the subset of self-homotopy equivalences. The subset inducing the trivial map on homotopy is known to be a nilpotent semigroup. We have determined the nilpotency of these semigroups in the case where spaces are rank 2 Lie groups and simply connected Hopf spaces. Moreover these semigroups define a filtration on a self … More -homotopy set. It is known that the filtration has finite length. We defined numerical invariants sz(X) and lz(X) for a space X. Then we have determined these two numbers for compact Lie groups. Next we summarize our results on self-homotpy equivalences. For any space the homotopy classes of self-homotopy equivalences is a group. It is neither abelian nor nilpotent in general. However subgroups associated with homotopy groups which are defined similarly as for the above case are known to be nilpotent groups. Nilpotent groups have nice properties like abelian groups. In particular there exists localization theory for nilpotent groups. We can transfer some results obtained for the sets which consists of maps inducing the trivial map on homotopy to self-homotopy equivalences group by using localization theory.
Through this study we have obtained the results above. Further, we have realized some new important problems in this project which we will study in future.
The investigators have contributed to the project by their considerations from their special fields. Less

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (27 results)

All 2005 2004 2003 Other

All Journal Article (15 results) Publications (12 results)

  • [Journal Article] The construction of units of infinite order in the character ring of a finite group2005

    • Author(s)
      山内 憲一
    • Journal Title

      Yokohama Mathematical Journal 51

      Pages: 89-97

    • NAID

      120001740808

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] The construction of units of infinite order in the character ring of a finite group2005

    • Author(s)
      Kenichi Yamauchi
    • Journal Title

      Yokohama Mathematical Journal Vol51

      Pages: 89-97

    • NAID

      120001740808

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The nilpotency of self-homotopy sets2004

    • Author(s)
      丸山 研一
    • Journal Title

      Proceedings of the international conference on homotopy theory and related topics, Korea University

      Pages: 39-44

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] The nilpotency of self-homotopy sets2004

    • Author(s)
      Ken-ichi Maruyama
    • Journal Title

      Proceedings of the international conference on homotopy theory and related topics, Institute of Science and Technology(Korea University)

      Pages: 39-44

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Web教材についての考察2004

    • Author(s)
      越川 浩明
    • Journal Title

      千葉大学教育学部研究紀要 52

      Pages: 307-311

    • NAID

      110004628713

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Effectiveness of a Web Bulletin Board Enhanced with a Knowledge Map2003

    • Author(s)
      越川浩明(第一著者 永井正洋)
    • Journal Title

      Educ.Technol.Res 26

      Pages: 41-52

    • NAID

      110002982161

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Jacobson radical of the character ring of a finite group2003

    • Author(s)
      山内 憲一
    • Journal Title

      千葉大学教育学部紀要 51

      Pages: 315-317

    • NAID

      110004628661

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The Effectiveness of a Web Bulletin Board Enhanced with a Knowledge Map2003

    • Author(s)
      Hiroaki Koshikawa(with M.Nagai et al.)
    • Journal Title

      Educ.Technol.Res. 26

      Pages: 41-52

    • NAID

      110002982161

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On the Jacobson radical of the character ring of a finite group2003

    • Author(s)
      Kenichi Yamauchi
    • Journal Title

      Bulletin of The Faculty of Education(Chiba University) Vol.51

      Pages: 315-317

    • NAID

      110004628661

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] π*-kernels of Lie groups

    • Author(s)
      丸山 研一
    • Journal Title

      Transaction of American Math.Society(掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Determination of multiplicative nilpotency of homotopy sets

    • Author(s)
      丸山 研一
    • Journal Title

      Geometry and Topology Monographs(掲載予定)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] π^*-kernels of Lie groups

    • Author(s)
      Ken-ichi Maruyama
    • Journal Title

      Transaction of American Math.Society (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Determination of the multiplicative Nilpotency of self-homotopy sets

    • Author(s)
      Ken-ichi Maruyama
    • Journal Title

      Geometry and Topology Monographs (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] π*-kernel of Lie groups

    • Author(s)
      丸山 研一
    • Journal Title

      Transactions of the American Mathematical Society (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Determination of the multiplicative nilpotency of seif-homotopy sets

    • Author(s)
      丸山 研一
    • Journal Title

      Geometry and Topology (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Publications] 丸山 研一: "π_*-kernels of Lie groups"Transactions of the American Mathematical Society. (発表予定).

    • Related Report
      2003 Annual Research Report
  • [Publications] 丸山 研一: "Stability properties of maps between Hopf spaces"Quarterly Journal of Mathematics, Oxford Ser(2). 53. 47-57 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] 丸山 研一(第一著者M.Arkowitz): "The semigroup of self-homotopy classes which induce zero on homotopy groups"Kyushu Journal of Mathematics. 56. 89-107 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] 越川 浩明(第一著者 永井正洋): "Web上での協同学習の携帯電話を用いた形成的評価に関する研究"日本科学教育学会年会論文集. 27. 189-192 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 越川 浩明: "Web教材についての考察"千葉大学教育学部研究紀要. 152(発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] 山内 憲一: "On the Jacobson radical of the character ring of a finite group"千葉大学教育学部研究紀要. 51. 315-317 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 丸山 研一: "Stability properties of maps between Hopf spaces"Quarterly Journal of Mathematics. 53. 47-57 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 丸山 研一(第一著者 M.Arkowitz): "The semigroup of self-homotopy classes which induce zero on homotopy groups"Kyushu Journal of Mathematics. 56. 89-107 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 丸山 研一: "A subgroup of self homotopy equivalences which is invariant on genus"Contemporary Mathematics. 274. 225-231 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 山内 憲一: "On the Jacobson radical of the character ring of a finite group"Bulletin of the Faculty of Education Chiba University. 51. 315-317 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 越川 浩明(第一著者 永井正洋): "知識マップ上での協同学習における携帯端末を用いた形成的評価に関する研究"日本教育工学会第18回全国大会講演論文集. 393-394 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 越川 浩明(第一著者 永井正洋): "The Effectiveness of a Web Bulletin Board Enhanced with a Knowledge Map"ICCE2002 in Auckland, New Zealand, Proceedings. 1. 268-272 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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